finite element analysis of the dish multi-point forming …...finite element analysis of the dish...

Finite Element Analysis of the Dish Multi-Point

Forming Process

Prof. Dr. Hani Aziz Ameen*

Asst. Prof. Dr. Haidar Akram Alsabti*

Zahraa Abdul Kareem Radhi* *Middle Technical University

Technical Engineering College

Baghdad

Abstract - This study focuses on simulation of multi-point die to produce the dish uses Bezier function for surface modeling to get the

best modeling points for dish. The mathematical functions required to generate curves and surfaces are investigated together with

complete algorithms using MatLap program for their achievement. Finite element method software (ANSYS 15.0) was used to

simulate the dish. A multi-point die was built with the same dimensions suggested by Wang and Li [1], quarter of the die in three

dimensional model is used due to symmetry of the required product. Copper plate has been chosen with 5KN blank holder force,

three rubber’s thickness (elastic cushion) are used (0.5,0.7,2)mm to avoid dimples also three radii of punch group (3,4,6)mm are

investigated. It is concluded that the punch element with radius 4mm and elastic cushion (rubber) 2mm is the best design to avoid the

dimples defect and have the homogenously distributed stress.

Keywords: Multi-point forming, Bezier surface, Dish, MATLAB program, control points, Finite Element Method, ANSYS, Rubber, Stress

analysis.

1. INTRODUCTION

Multi-point forming (MPF) is a flexible forming technology in which the fixed shape of conventional dies is replaced via

movable elements call “punch group”. Any complex design need to represent its lines ,curves and surface mathematically. the

accurate design of any cimplex part depending on the curves &surfaces and its degree of the curve equation. The most used

surfaces to describe the complex parts are the third degree surfaces because of their high accuracy, but the disadvantage of using

this degree of surfaces is its limitation to represent the desired parts [2],[3]. The curves that caused using Beziers method are

called Bezier curves [4]. The Bézier curve, referred to the French researcher Pierre Bézier. The Bézier surface is the extended

of the Bezier curve. It is consist of dragging Bezier curved via spaced to generate the [5]. Analyzed the deformation mechanism

of the multi-point dieless forming and positioning technology with commercial FEM codes [6]. Bezier surface was used to the

surface of Dish in (3D) model by using MATLAB software to determine the coordinate of all points along curve and locate

them on the surface[7],[8]. Qian et al.,[9] (2007) studied the analysis of multi-point forming for dish head and concluded that

the forming quality is improved by MPF. Wardhani et al., [10] (2014) studied the numerical simulation of multipoint forming

used two model the first one was pins of die MPF arranged in hexagonal packing and the other one was squared and found that

the hexagonal packing is more efficient. Zhong. et al.,[11] described approach the incremental displacement on the basis of

Lagrangian formulation and elastio–plastic material model for finite element method of MPF process. analyzed MPF by using a

new finite element code. The results of numerical shown good agreement with those of the experiments and simulation.

In this study MATLAB software program, is used to find the coordinate of all dish points that are situated on the Bezier curve

and surface to create the forming dish surface by adjusting the height of the every punch by ANSYS according to resulted

point's coordinates of the MatLap software and different parameters of punch group and rubber thickness are used to find the

optimum case.

2. MATHEMATICAL REPRESENTATION OF BEZIER SURFACES

The Bezier surface is a direct extension of a Bezier curve. The simple extension for three dimensional free–form surfaces from

3-dimensional free-form curve is by incorporating another parameter (s) to the vector equation of the curve to obtain the surface

equation:

P(t,s) = [x(t,s) y(t,s) z(t,s)] …(1)

where: 0 ≤ (t,s) ≤ 1 and they are independent variables. Bezier surface is an extended of a Bezier curve[12]:

𝑃(𝑡, 𝑠) = ∑ ∑ 𝑏𝑖𝑗𝐵𝑖,𝑛(𝑡)𝐵𝑗,𝑚(𝑠) …… . . (2 )𝑚𝑗=0

𝑛𝑖=0

0 ≤ (t,s) ≤ 1 ; i,j = 0 ,…., n,m.

The bi j rectangular array of control points, Bi,n(t) and Bj,m(s) are the basis functions, defined in the same way as for "Bezier

curves". The general equation of a Bezier surface is [13]:

𝑃(𝑡, 𝑠) = 𝑇1∗𝑛 𝑀𝐵,𝑛∗𝑛 𝑏𝑛∗𝑚 𝑀𝐵 ,𝑚∗𝑚 𝑇 𝑆𝐵 ,𝑚∗1

𝑇 … . . (3 ) where the size of the matrices depend on the dimensions of the control point array defined by a (4x4) array of control points.

[2].

𝑃(𝑡, 𝑠) = 𝑇1∗4 𝑀𝐵,4∗4 𝑏4∗4 𝑀𝐵 ,4∗4 𝑇 𝑆𝐵 ,4∗1

𝑇 (4 )

International Journal of Engineering Research & Technology (IJERT)

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IJERTV5IS070269(This work is licensed under a Creative Commons Attribution 4.0 International License.)

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That the subscripts of the matrices represent their dimensions. Expanding equation (3) to get:

𝑃(𝑡, 𝑠) = [(1 − 𝑡)3 3𝑡(1 − 𝑡)2 3𝑡2(1 − 𝑡) 𝑡3] ∗ [𝑏] ∗

[

(1 − 𝑠)3

3𝑠(1 − 𝑠)2

3𝑠2(1 − 𝑠)

𝑠3 ]

…… . . . . ( 5 )

𝑃(𝑡, 𝑠) = [𝑡3 𝑡2 𝑡 1] ∗ [

−1 3 − 3 1 3 − 6 3 0−3 3 0 01 0 0 0

] ∗

[ 𝑏11 𝑏12 𝑏13 𝑏14𝑏21 𝑏22 𝑏23 𝑏24𝑏31 𝑏32 𝑏33 𝑏34𝑏41 𝑏42 𝑏43 𝑏44

]

∗ [

−1 3 − 3 1 3 − 6 3 0−3 3 0 01 0 0 0

] ∗ [

𝑠3

𝑠2

𝑠1

] … … …… … . . (6)

3. SURFACE MODEL IN THE SIMULATION PROCESS

The mathematical formulation of Bezier techniques have been achieved and programmed with MATLAB program to create the

curves and surfaces depending on control points. only one quarter of (3D) model have been established to generate the dish as

shown in Fig.(1) [14].

Fig.(1) control points of the dish

The control points of the surface represented the Bezier surface with 4x4 control points are:

(100,0,0)14(86,50, 0) b13(50,86.6, 0) b12b (0,100, 0) 11b

45)-(89, 0,2445) b-(89,0,2345) b-(44,77, 2245) b-(0,89,21b

80)-(58,0,3480) b-(50,29,3380) b-(29,50,3280) b-(0,58,31b

98)-(15, 0,44b 98) -(13,7,4398) b-(7,13,4298) b-(0,15,41b

Fig.(2) shown the Bezier surface for the quarter dish using MatLap software .

Fig.(2) Representation the quarter composite Bezier Surface for dish

3.1 MATLAB Program to Generate Dish

The proposed algorithm was coded in MATLAB Ver. (6.5) . Several bi-cubic Bezier patches were designed and tested to

generate the dish .

clear;

"clf;

"u=0:0.04:1;"

"for i=1:25"




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"U(i,1)=u(i)^3;"

"U(i,2)=u(i)^2;"

"U(i,3)=u(i);"

"U(i,4)=1;"

"end"

"w=0:0.04:1;"

"for i=1:25"

"W(i,1)=w(i)^3;"

"W(i,2)=w(i)^2;"

"W(i,3)=w(i);"

"W(i,4)=1;"

"end"

M=[-1 3 -3 1; 3 -6 3 0; -3 3 0 0; 1 0 0 0;];

CVX=[0,50,86.6025,100;0,44.5503,77.1634,89.1007;0,29.3893,50.9037,58.7785;0,7.8217,13.5476,15.6434;];

CVY=[100,86.6025,50,0;89.1007,77.1634,44.5503,0;58.7785,50.9037,29.3893,0;15.6434,13.5476,7.8217,0;];

CVZ=[0,0,0,0;-13.6197,-13.6197,-13.6197,-13.6197;-24.27051,-24.27051,-24.27051,-24.27051;-29.63064,-29.63064,-

29.63064,-29.63064;];

for i=1:25

for j=1:25

PX(i,j)=U(i,:)*M*CVX*M'*W(j,:)';

"PY(i,j)=U(i,:)*M*CVY*M'*W(j,:)';"

"PZ(i,j)=U(i,:)*M*CVZ*M'*W(j,:)';"

"end"

"end"

surf(PX,PY,PZ);grid on;

x=PX

y=PY

z=PZ

4. FINITE ELEMENT ANALYSIS(FEA)

The finite element Analysis is the numerical method too for predicting the stress-strain etc… of an engineering problem. Later,

it was extended to predicting the behavior of complex structure such as to obtain the pressure field and velocity of fluid flow as

well as the temperature. The Exact solution of the complex problem had complicated shapes of loadings are difficult and hard to

derive and may be impossible to obtain. So FEM is used to do that.

5. NUMERICAL SIMULATION

Because of symmetry, only one half of the real sector is modeled and the following assumptions are used in the analysis

presented in this paper:

1. During the process, the plate temperature remains constant. 2. Thermal expansion is neglected. 3. The strain rate has no effect on material stress – strain relation. 4. The material has a bilinear total stress - total strain curve. 5. Pins, Dies and Blank holder are rigid bodies. 6. The friction coefficients are constant

The following steps the general procedure to simulate the problem in "ANSYS code ":

5.1SELECATION OF THE ELEMENT TYPE

1. Forming process by using Multi-point die, the finite element model is consist of five parts: upper discrete pin punches,

lower discrete pin punches, plate(blank), blank holder and die. For (3D model). Pins, die and the blank holder are represented as

a rigid body and the plate is meshed with (Solid element 185). The contact interface between the die and the blank is

represented by (contact element 172 and target 170) [16].

2. Forming process by using Multi-point die (with using elastic cushion), the finite element model is consist of seven parts:

upper discrete punches, lower discrete punches, plate, upper (elastic cushion) layer, lower (elastic cushion) layer, blank holder

and die. Pins, die and the blank holder are represented as a rigid and the elastic cushion material is modeled with (Hyperelastic

Solid element 185). The contact interface between the die and the deformed material is represented by (contact element 172 and

target 170).

5.2 Material properties

Copper is the proposed material for analysis, the mechanical properties of materials as shown in table (1) .




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Table (1) Mechanical properties of the copper metal

Property Copper (Cu)

Modulus of elastic (E) 124 Gpa

Tangent modulus(ET) 0.8 Gpa

Yield stress (σy) 45 Mpa

Poissonʼs ratio(υ) 0.34

5.3 Hyperelastic Material Model

In 1951,Rivlin and Sunders developed a hyperelastic material model for large deformations of rubber. This material model is

assumed to be incompressible and initially isotropic.

The form of strain energy potential for a Mooney-Rivlin material is given as[17] :

W= 𝑐10(𝐼1 − 3) + 𝑐01(𝐼2 − 3) + 1/𝑑(𝐽 − 1)2 ………..(7)

Where c10 , c01 and d are material constants.

Mooney-Rivlin (2 parameters) is used to represent the elastic cushion . The hyperelastic material model to describe the

behavior of the elastic cushion. In the following analyses, the friction coefficient is assumed as (0.1).

Property Elastic cushion

Modulus of elastic (E) 2.87 Mpa

Poissonʼs ratio(υ) 0.499

Mooney-Rivlin Constants

c10 0.293 Mpa

c01 0.177 Mpa

6. Cases Studies

Three radii of pin punches are used to observed the optimum radius of pin punch used in the MPF process.

A- Radius of pin punch = 4mm

The effect of radius punch is various by various thickness of cushion on dimple suppression, are the using without rubber

,(0.5mm) rubber thickness, (0.7mm) rubber thickness and (2mm) rubber thickness elastic cushion.

1-Without Rubber

Firstly no rubber as cushion is used, the pin punches contact directly with blank that caused wrinkling and dimpling which are

the most defects in multi- point forming(MPF) of sheet metal. Those phenomenon affect the shape quality. Dimple happens due

to discontinuous contact between the sheet and pin punches. As shown in Figure (1).

Upper punch

Blank holder lower punch

Copper blank

(a)model of MPF (b) 3D quarter of the model




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(c) displacement in the dish (d) equivalent Strain

(e) Equivalent Stress (f) Equivalent Stress

(g) Equivalent Stress (side view) for Dish (h)Equivalent Stress (quarter dish)

Figure(1) the deflection, equivalent strain , equivalent stress of dish without rubber

2- Blank cover from top and bottom by (0.5mm) thickness Rubber.

When using the rubber with thickness 0.5 mm as a cover to the forming area ,the result illustrated the appearance of the

smallest dimple. The pin

punch causes the concentrated force on the small contact area between the metal plate, therefore the

plastic deformation are caused , that called "dimple"

as shown in Figure 2

(a) Model of dish (b) displacement




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(c) Equivalent Stress (d) Equivalent Stress

Figure(2) the deflection, equivalent strain , equivalent stress of dish with 0.5mm rubber

3- Blank cover from top and bottom by 0.7mm thickness Rubber.

Used rubber with thickness 0.7 mm upper and lower plate and the result showed the wrinkling are least as before, as shown in

figure (3)

Figure(3) the deflection, equivalent strain , equivalent stress of dish with 0.7mm rubber

4-Blank cover from top and bottom by 2mm thickness Rubber.

When used the thickness of rubber 2mm upper and lower plate the result illustrated that the surface of the dish is free wrinkling

and high surface quality. Figure (4) observed the results of 2mm thickness of rubber.

(c) equivalent Stress (d) Equivalent Stress

(a) equivalent stress (b) equivalent strain




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(c)

Figure(4) the deflection, equivalent strain , equivalent stress of dish with 2mm rubber

B- Radius of pin punch = 3mm

In this case the radius of pin punch is taken 3mm, and the thickness of the rubber is 2mm, as a result of the previous case, which

can be concluded that 2mm is the best rubber thickness. Figure(5) shown the results.

displacement (d) equivalent stress (side view)

(e) equivalent stress (f) complete model

(a) stress equivalent (b) stress equivalent

(a) model (b) equivalent stress




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(c) stress equivalent (d) stress equivalent


C- Radius of pin punch = 6mm

The other case when the end of hemispherical end is (6mm) ,the changing is more stable , and these observed the dimples are

smaller on the formed part .From the result ,it can be concluded that the larger end radius of punch element is given smaller

dimples. Figure (6) shown the results.

(a) model (b) equivalent strain

(c) equivalent stress (d) displacement


When the punch radius increasing the contact area is increased with reducing the curvature , and dimple will be smaller .

The ANSYS results shown that the larger hemispherical end radius should be selected to insure that the contact between the

blank and punch element is good.

But in our case ( production dish via MPF) , the results shown that the pin punches radius (4mm) product the dish without

dimple. So it can be deduced that the 4mm radius is the optimum radius for MPF dish with 2mm thickness rubber.

7. FORMING LOAD – DISPLACEMENT RELATION

The punches force – displacement curves of the dish during MPF are determined in this paper for all cases done and it seem that

load increased with the displacement firstly then decreasing when the material goes to plastic deformation as shown in

Figure(7).




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Figure(7) punch force – displacement curve with different punch pin raduii ( the force in KN- Displacement in mm)

8. CONCLUSIONS

This paper is to develop an algorithm that generate the surface of dish. A designed surface is represented by control points the

surface has been represented depending on Bezier curve to generate the dish surface. The control points generation for a disk is

developed. Applying Bezier surface controls points for an existing set of collected three dimensional data points is introduced

to describe such dish. The proposed algorithm for dish generation was developed and implemented successfully through

ANSYS software. Copper plate has been chosen with three rubber’s thickness (elastic cushion) (0.5,0.7,2)mm to avoid dimples

also three radii of punch group (3,4,6)mm are investigated. It is concluded that the punch element with radius 4mm and elastic

cushion (rubber) 2mm is the best design to avoid the dimples defect and have the homogenously distributed stress.

9. REFERENCES

[1] Wang L., and Li L., “Computer Simulation of Thick Plate Multi-point Forming Process”, International Conference on computer design and applications, Vol.3, pp.143-146, (2010).

[2] Akeel S. B. , "Automatic Surface Generation from Wireframe Data in CAD Applications", M.Sc. Thesis, Department of Production and Metallurgy Eng., University of Technology, Baghdad , Iraq ,(2006).

[3] Nicholas M. P. and Takashi M. "Shape Interrogation for Computer Aided Design and Manufacturing" Springer-Verlag Berlin Heidelberg (2002). [4] Ali K. K. and Emad A. N."Computer-Based Design And Manufacturing" Springer Science and Business Media, LLC (2007). [5] Kimura, F., Yamaguch, Y., Koboyashi, K, Nakajima, H.," Free form surfaces modelling for complicated product description ", Annual of the CIRP Vol.39

/1/ 1990.

[6] Liu Chunguo, Li Mingzhe, Cai Zhongyi, Yan Aimin , Su Shizhong, Digitally Adjustable Tooling Technology for Manufacturing of Aircraft Panels, 2010 International Conference on Digital Manufacturing & AutomationR.

[7] Mustafa E. A., “Computer Aided Design and Analysis of Reconfigurable Discrete Dies for Sheet Metal Forming” , M.Sc. Thesis, Department of Mechanical Engineering (Applied Mechanics) , University of Babylon , Iraq,(2013).

[8] Ragad A. N., “Numerical and Experimental Investigation of Multi Points Die Forming” , M.Sc. Thesis, Department of Mechanical Engineering (Applied Mechanics) , University of Babylon , Iraq,(2013).

[9] Z.R Qian.,F.X.Tan.," The analyse on the process of multi-point forming for dish head" journal of materials processing technology vol.187-188(2007). [10] Wardhani, Rivai .et al."Numerical Simulation of Multipoint Forming with Circular Die Pins in Hexagonal Packing" Applied Mechanics and Materials Vol

493 (2014) pp 589-593.

[11] Zhong-Yi C. and Ming-Zhe L., "Finite Element Simulation of Multi Point Sheet Forming Process Based on Implicit Scheme" J Mater Process Technol, Vol. 161, pp. 449–455, (2005).

[12] Shi-Min H., Rou-Feng T., Tao J. and Jia-Guang S., "Approximate merging of a Pair of Bezier Curves" Computer-Aided Design, Vol 33, pp 125-136,( 2000).




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[13] Andres I., "Bezier Curves And Surfaces" University of Cantabria, Santander, Spain, 2001. [14] Dale A. Patrick , ”Bezier Surface Generation of the Patella”, M.Sc. thesis, Wright State [15] Nakasone Y. and Yoshimoto S.," Engineering Analysis With ANSYS Software", Department of Mechanical Engineering, Tokyo University of Science,

2006.

[16] Ansys 15 User Guide . ( 2015).

[17] Kamal Jahani and Hossein Mahmoodzade , “Predicting the dynamic material constants of Mooney-Rivlin model in broad frequency range for elastomeric components” Latin American Journal of Solids and Structures , 11 , pp 1983-1998, (2014).




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finite element analysis of the dish multi-point forming …...finite element analysis of the dish...

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